Problem $7$:

Consider the system shown at its

equilibrium position given in the figure.

$AB$ is a rigid rod with mass $m,$ length

$L=l_1+l_2+l_3,$ and centroidal radius

of gyration $k_G.$ Links $AC$ and $BD$ are

massless.

$($a$)$ Find the equivalent single$-$degree$-$

of$-$ freedom system in the form

$m_{eq}{x}_1^{}+c_{eq}{x}_1^{}+k_{eq}{x}_1=0$

using the parameters given in the figure

without the numerical values.

Now, assume $m=1kg{m}_1=10kg,$

$m_2=5kg,k_G=0\mathrm{.}1m,k_1=100\frac{N}{?}$

$m,k_2=150\frac{N}{m},l_1=1\mathrm{.}2m,l_2=$

$0\mathrm{.}3m,l_3=1\mathrm{.}5m.$

$($b$)$ What should be the value of damping coefficient, $c,$ to have the system critically damped?

If $m_2$ is initially displaced by $x_2(0)=0\mathrm{.}01m$ and the system is released from rest, what will

be the maximum speed of $m_2$ during its motion?

$($c$)$ What should be the damping coefficient, $c,$ for this system if the amplitude ratio between

two successive cycles is $0\mathrm{.}25?$ What will be the period of these oscillations?